Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W^⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V.

General bilinear forms

Let $V$ be a vector space over a field $F$ equipped with a bilinear form $B$. We define $u$ to be left-orthogonal to $v$, and $v$ to be right-orthogonal to $u$, when $B(u,v)=0$. For a subset $W$ of $V$ we define the left orthogonal complement $W^\bot$ to be

$W^\bot=\left\{x\in V : B( x, y ) = 0 \mbox{ for all } y\in W \right\}\,.$

There is a corresponding definition of right orthogonal complement. For a reflexive bilinear form, where $B(u,v)=0$ implies $B(v,u)=0$ for all $u$ and $v$ in $V$, the left and right complements coincide. This will be the case if $B$ is a symmetric or an alternating form.

The definition extends to a bilinear form on a free module over a commutative ring, and to a sesquilinear form extended to include any free module over a commutative ring with conjugation.

Properties

• An orthogonal complement is a subspace of $V$;
• If $X\subset Y$ then $X^\bot \supset Y^\bot$;
• The radical $V^\bot$ of $V$ is a subspace of every orthogonal complement;
• $W\subset (W^\bot)^\bot$;
• If $B$ is non-degenerate and $V$ is finite-dimensional, then $\dim(W)+\dim (W^\bot)=\dim V$.

Example

In special relativity the orthogonal complement is used to determine the simultaneous hyperplane at a point of a world line. The bilinear form η used in Minkowski space determines a pseudo-Euclidean space of events. The origin and all events on the light cone are self-orthogonal. When a time event and a space event evaluate to zero under the bilinear form, then they are hyperbolic-orthogonal. This terminology stems from the use of two conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal.

Inner product spaces

This section considers orthogonal complements in inner product spaces.

Properties

The orthogonal complement is always closed in the metric topology. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensional Hilbert spaces, some subspaces are not closed, but all orthogonal complements are closed. In such spaces, the orthogonal complement of the orthogonal complement of $W$ is the closure of $W$, i.e.,

$(W^\bot)^\bot = \overline W$.

Some other useful properties that always hold are the following. Let $H$ be a Hilbert space and let $X$ and $Y$ be its linear subspaces. Then:

• $X^\bot = \overline X^\bot$;
• if $Y \subset X$, then $X^\bot \subset Y ^\bot$;
• $X\cap X^\bot = \{0\}$;
• $X\subset (X^\bot)^\bot$;
• if $X$ is a closed linear subspace of $H$, then $(X^\bot)^\bot = X$;
• if $X$ is a closed linear subspace of $H$, then $H = X \oplus X^\bot$, the (inner) direct sum.

The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span.

Finite dimensions

For a finite-dimensional inner product space of dimension n, the orthogonal complement of a k-dimensional subspace is an (n − k)-dimensional subspace, and the double orthogonal complement is the original subspace:

(W^⊥)^⊥ = W.

If A is an m × n matrix, where Row A, Col A, and Null A refer to the row space, column space, and null space of A (respectively), we have

(Row A)^⊥ = Null A

(Col A)^⊥ = Null A^T.

Banach spaces

There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complement of W to be a subspace of the dual of V defined similarly as the annihilator

$W^\bot = \left\{\,x\in V^* : \forall y\in W, x(y) = 0 \, \right\}.\,$

It is always a closed subspace of V^∗. There is also an analog of the double complement property. W^⊥⊥ is now a subspace of V^∗∗ (which is not identical to V). However, the reflexive spaces have a natural isomorphism i between V and V^∗∗. In this case we have

$i\overline{W} = W^{\bot\,\bot}.$

This is a rather straightforward consequence of the Hahn–Banach theorem.

See also

• Complemented lattice

External links

• Instructional video describing orthogonal complements (Khan Academy)